Dynamical origin of the hidden superstring sectors of the critical 26-dimensional bosonic string

The consistent ten-dimensional closed superstrings are hidden, as tachyon-free space-time-supersymmetric subspaces, in the Hilbert space of the critical 26-dimensional bosonic closed string: after compactification on the $E_8\times SO(16)$ lattice, a spectral truncation — discarding all oscillator states of twelve compactified bosons and keeping only their zero modes — reproduces exactly the two $N=1$ heterotic strings and the two $N=2$ type-II strings. This "hidden superstrings" result (Englert, 1986) is purely kinematical, and the problem it leaves open is dynamical: no mechanism is known by which the bosonic string would select and occupy one of its truncated supersymmetric sectors rather than its tachyonic vacuum. The problem is to give a non-perturbative, beyond-S-matrix account of the critical bosonic string that (i) explains the selection among the allowed self-dual compactifications and compatible truncations, (ii) dynamically realizes the decoupling of the twelve compactified bosonic degrees of freedom so that space-time fermions, world-sheet superghosts, and supersymmetry emerge while the tachyonic vacuum is exited, and (iii) determines the stability of, and transitions between, the tachyonic and supersymmetric phases. The only proposed mechanism, condensation of string-field expectation values (1986), was never developed; 2001–2008 work supplies circumstantial evidence only.

Background

Perturbative string theory describes relativistic strings moving in a space-time (the "target space") whose critical dimension is $D=26$ for the bosonic string — the dimension at which the quantum conformal (Virasoro) anomaly of the two-dimensional world-sheet theory cancels. The closed bosonic string is normally studied as a perturbative S-matrix theory, that is, as a rule for computing scattering amplitudes expanded around a pre-selected background, and it has two famous defects: its spectrum contains no space-time fermions, and it contains a tachyon, a mode of imaginary mass signaling that the chosen vacuum is unstable.

Two classical mechanisms are the ingredients of the question. First, toroidal compactification: rolling up $d$ of the 26 coordinates on a torus whose allowed momenta lie on an even self-dual lattice (a lattice of points with integer squared lengths that coincides with its own dual) leaves $26-d$ macroscopic dimensions; when the lattice is the weight lattice of a simply laced group $G$ (all roots of equal length), the massless states with lattice momenta become gauge bosons of $G$ — the Frenkel–Kac mechanism by which gauge symmetry arises from compactification geometry. Second, in light-cone gauge — a choice of world-sheet coordinates that keeps only the physically propagating transverse oscillators — the string has $24$ transverse coordinates; compactifying $d=24-s$ of them leaves an internal symmetry $SO_{\mathrm{int}}(s)$, and states in spinor representations of $SO_{\mathrm{int}}(s)$ transform as space-time fermions once the transverse space-time rotation group is identified with the diagonal of $SO_{\mathrm{trans}}(s)\times SO_{\mathrm{int}}(s)$ — the string-theoretic analogue of the Jackiw–Rebbi mechanism by which fermionic states can emerge from bosonic fields in a topologically non-trivial background.

In 1985–86, Casher, Englert, Nicolai and Taormina, and then Englert, Nicolai and Schellekens, combined these observations with a spectral truncation. Counting world-sheet central charges (the anomaly coefficients of the two-dimensional conformal symmetry) shows that accommodating local world-sheet supersymmetry costs the equivalent of twelve compactified bosons: the superghost system of the covariant superstring (the ghost fields that implement local world-sheet supersymmetry) contributes central charge $11$, and the unphysical longitudinal and time-like Majorana fermions contribute $2\times\frac{1}{2}$. One therefore truncates — discards — all states created by oscillators of twelve of the compactified bosons, keeping only their zero modes (their momentum and winding states, which carry no oscillator excitation); consistency then requires $s\le 8$, so the highest space-time dimension in which fermions can appear is $10$. For $s=8$ the compactification lattice is $E_8\times SO(16)$, with $SO(16)$ decomposed as $SO'(8)\times SO(8)$ and the truncated oscillators those of the $E_8\times SO'(8)$ root lattice. The truncated Hilbert space then reproduces exactly the four consistent ten-dimensional closed superstrings — the two $N=1$ heterotic strings and the two $N=2$ type-II strings — tachyon-free and space-time supersymmetric. Englert's 1986 review "Hidden Superstrings" summarized the conclusion that all consistent closed ten-dimensional superstrings are hidden in Poincaré supersymmetric, tachyon-free sectors of the 26-dimensional bosonic closed string. Later work (2001–03) extended the truncation to open strings: imposing the absence of massless tadpoles (linearized divergences in one-loop vacuum amplitudes) on the bosonic theory compactified on $E_8\times SO(16)$ selects the Chan–Paton gauge group (the group whose indices label open-string endpoints) of the type-I superstring, and predicts the tensions of fermionic D-branes (hypersurfaces on which open strings end) — circumstantial evidence that the truncation "knows about" the fermionic strings.

Everything above is kinematical: it identifies consistent subspaces of the bosonic string's Hilbert space and checks their internal consistency. The construction is silent on dynamics. A perturbative S-matrix formulation cannot say which vacuum the theory actually occupies, whether a hand-imposed spectral truncation can be reached by dynamical evolution, or how the bosonic string escapes its tachyonic instability. The only mechanism ever proposed — expectation values of string fields (the second-quantized wave-functional formulation of string theory) condensing so as to move the theory from the tachyonic vacuum to a truncated vacuum — was put forward by Englert, Nicolai and Schellekens in 1986 and, by the program's own later assessment, "remained a pure speculation devoid of any predictive content". Answering the question requires a non-perturbative, beyond-S-matrix formulation of the critical bosonic string, which does not presently exist.

Problem Statement

Provide a dynamical account, within a non-perturbative, beyond-S-matrix formulation of the critical ($D=26$) closed bosonic string, of the truncated tachyon-free space-time-supersymmetric sectors discovered in the hidden-superstrings construction — the kinematical result that, after compactification of the bosonic string on the even self-dual $E_8\times SO(16)$ lattice, discarding all states created by oscillators of twelve of the compactified bosons while keeping their zero modes (their non-oscillator momentum and winding states) reproduces exactly the four consistent ten-dimensional closed superstrings. The account must rigorously establish, within one coherent framework: (i) vacuum selection — why and how the theory selects, or fails to select, among the allowed self-dual toroidal compactifications and their compatible spectral truncations, including in particular the $E_8\times SO(16)$ compactification on which the open-string tadpole-cancellation condition (the absence of massless tadpoles, i.e., of linearized divergences in the one-loop open-string vacuum amplitudes) is satisfied; (ii) dynamical decoupling — the mechanism by which the oscillator states of the twelve compactified bosonic fields decouple (their zero modes surviving), so that space-time fermions, the world-sheet superghost structure (the ghost fields of local world-sheet supersymmetry), and space-time supersymmetry emerge in the effective theory while the original tachyonic vacuum is exited; and (iii) phase structure — the stability of the resulting supersymmetric phases and the characterization of transitions between the conventional tachyonic phase of the bosonic string and the supersymmetric phases. A rigorous demonstration that no such dynamical mechanism exists — that is, that the truncated sectors are kinematical subspaces with no dynamical significance for the bosonic string's actual vacuum structure — resolves the problem equally.

References

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